Preview of the Practical Applications of the Tools
The Five Platonic Solids as Nesting Dolls (Containers)
This section is a preview of some of the information in the other portals. A principle theme of Newtools geometry and naturalmodular building techniques is the interconnectedness of the useful information embedded in the fundamental structures of basic geometry. One method of demonstrating this concept is thinking of geometrical systems (polyhedra) as containers, with the ability of fit exactly inside each other.
An introductory video treats the Five Platonic solids as 'nesting dolls'. Nesting dolls are a toy where one large doll opens to reveal a slightly smaller doll inside that in turn has a smaller doll inside, on till only one small doll left.
Containment of geometrical systems is based on numerical constants that determine size. Here we have the icosahedron with a dodecahedron inside, with a cube inside that, a tetrahedron inside the cube and an octahedron inside the tetra.
I found these constants using the You/Me BB measuring system (Basic Tool Kit portal). This new system lets you see the genealogy of calculation in each mathematical notation. The clear material is called DuraLar and is very helpful in showing the interconnectedness (synergy) of geometrical systems (polyhedra).
Cube Inside Dodecahedron
GR = Golden Ratio
GR = 0.618033988749895...
Edge of dodeca known divide
Edge of cube known multiply
(number less than 1 math reversed)
The cube and dodecahedron has a deep connection. There is a simple symmetric modular piece that defines the volume difference between these two systems (polyhedra). The long axis of the pentagon will be the same length as the edge of the cube that fits exactly inside. The GR constant also defines the relationship between the long axis of the pentagon and its edge.
If you explore the ideas of this website you will find clusters of systems have interchangeable traits. The cube, tetrahedron and octahedron have many interconnections. The volume difference between the cube and the tetrahedron that fits inside is an asymmetrical tetrahedron (has four sides) which is 1/8 volume of the 6BB octa. Eight of these modular units define the volume of an octahedron. We will explore the other options in the Structural Interconnectedness portal.
There is always a half-size octahedron inside any tetrahedron. When stacked together these two systems fill all of the space without gaps. This is one of the fundamental jobs description for basic structures. Interconnectedness is the key to understanding this simple version of Design-in-Nature.
In Nature, there are some series of pattern growth that when they reach the end of their cycle have a slight 'and ah' pause and the cycle begins again on another scale. This is a simple fractal pattern.
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Newtools maps out the communication networks that connect the center of each system with their outside limits. This helps increase natural pattern recognition.
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When you think of geometrical systems (polyhedra) as containers (fitting inside each other) you are thinking in whole systems which helps in finding synergy among the basic geometry.
